Instructor: Dr. Gábor ELEK

Text:  handout (the  notes of the lecturer)  (and M. Reed, B. Simon, Methods of Modern Mathematical Physics, vol. I.: Functional Analysis. Academic Press, New York and London 1972)

Prerequisite: competence in calculus, linear algebra, some real analysis and interest in learning beautiful mathematics.

Topics:

Basic real analysis
-- Compact metric spaces
-- Complete spaces
-- A short introduction to  measure theory
-- Vitali's theorem on the existence of non-measurable set
-- Borel sets
 

Finitely additive measures
--Ultrafilters
--Ultralimits and their applications
-- The Banach-Tarski paradox
 

Dynamical systems, entropy and fractals
-- Recurrence, transitivity, van der Waerden theorem
-- The entropy
-- Fractal spaces, fractal dimension

Banach spaces
-- Basics
-- Hilbert spaces
-- Dual spaces
-- Banach-Steinhaus and the open mapping theorem
 

With lots of concrete examples and problems!!!

The course is recommended for students in the last undergraduate year. Students with interest in mathematical and theoretical physics are specially invited.